Stabilization Effect of Magnetic Fields on Two-Dimensional Compressible Current-Vortex Sheets

We analyze the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic. In the case when the magnitude of the magnetic field has no jump on the current-vortex sheets, we find a necessary and sufficient condition of linear stability for the rectilinear current-vortex sheets, showing that magnetic fields exert a stabilization effect on compressible vortex sheets. In addition, a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, occurs in a priori estimates of solutions to the linearized problem for a rectilinear current-vortex sheet, as the Kreiss–Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces. In this study, the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss–Lopatinskii condition may fail simultaneously, plays a crucial role in the a priori estimates.

[1]  A. Majda,et al.  The stability of multidimensional shock fronts , 1983 .

[2]  Andrew J. Majda,et al.  Nonlinear development of instabilities in supersonic vortex sheets I. The basic kink modes , 1987 .

[3]  H. Kreiss.  Initial boundary value problems for hyperbolic systems , 1970 .

[4]  Yuri Trakhinin,et al.  The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics , 2009 .

[5]  Jean-François Coulombel,et al.  NONLINEAR COMPRESSIBLE VORTEX SHEETS IN TWO SPACE DIMENSIONS , 2008 .

[6]  Yuri Trakhinin,et al.  Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis , 2005 .

[7]  Yoshikazu Giga,et al.  Nonlinear Partial Differential Equations , 2004 .

[8]  John W. Miles,et al.  On the disturbed motion of a plane vortex sheet , 1958, Journal of Fluid Mechanics.

[9]  J. Miles.  On the Reflection of Sound at an Interface of Relative Motion , 1957 .

[10]  Andrew J. Majda,et al.  Initial‐boundary value problems for hyperbolic equations with uniformly characteristic boundary , 1975 .

[11]  Gui-Qiang G. Chen,et al.  Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws , 2012 .

[12]  I. N. Sneddon,et al.  Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves , 1999 .

[13]  R. Phillips,et al.  Local boundary conditions for dissipative symmetric linear differential operators , 1960 .

[14]  A. Majda,et al.  The existence of multidimensional shock fronts , 1983 .

[15]  Jean-François Coulombel,et al.  On the transition to instability for compressible vortex sheets , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  P. Lax,et al.  Systems of conservation laws , 1960 .

[17]  D. Serre.  Systems of conservation laws , 1999 .

[18]  Gui-Qiang Chen,et al.  Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics , 2006 .